Find the Extreme Values of the Function Subject to the Given

Multiple Choice

Find the extreme values of the function subject to the given constraint.
- $$f ( x , y ) = y ^ { 2 } - x ^ { 2 } , \quad x ^ { 2 } + y ^ { 2 } = 16$$

A) Maximum: 32 at $( 0 , \pm 4 \sqrt { 2 } )$ ; minimum: $- 32$ at $( \pm 4 \sqrt { 2 } , 0 )$
B) Maximum: 16 at $( 0 , \pm 4 )$ ; minimum: $- 32$ at $( \pm 4 \sqrt { 2 } , 0 )$
C) Maximum: 32 at $( 0 , \pm 4 \sqrt { 2 } )$ ; minimum: $- 16$ at $( \pm 4,0 )$
D) Maximum: 16 at $( 0 , \pm 4 )$ ; minimum: $- 16$ at $( \pm 4,0 )$